Theorem pnrmtop 23714

Description: A perfectly normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
pnrmtop	⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Top)

Proof of Theorem pnrmtop

Step	Hyp	Ref	Expression
1		pnrmnrm 23713	. 2 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Nrm)
2		nrmtop 23709	. 2 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
3	1, 2	syl 18	1 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Top)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Topctop 23266  Nrmcnrm 23683  PNrmcpnrm 23685

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-cnv 5666  df-dm 5668  df-rn 5669  df-iota 6520  df-fv 6574  df-ov 7454  df-nrm 23690  df-pnrm 23692

This theorem is used by:  pnrmopn  23716